| Project/Area Number |
21K03197
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Sala Francesco 東京大学, カブリ数物連携宇宙研究機構, 客員准科学研究員 (60800555)
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| Project Period (FY) |
2021-04-01 – 2026-03-31
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| Project Status |
Granted (Fiscal Year 2022)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2025: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2024: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2023: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2022: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2021: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
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| Keywords | Hall algebras / COHAs / Categorification / Motivic Hall algebras / Quantum groups / Yangians / PT stable pairs / Space curve singularity / Quantum Groups / Hall Algebras / Higgs bundles / Flat bundles / stable pairs |
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| Outline of Research at the Start |
The project stands at the crossroads between representation theory and geometry. It aims at discovering new algebraic structures (Lie algebras associated with curves) which should encode geometric properties of (moduli) spaces appearing in the so-called Riemann-Hilbert and the non-abelian Hodge correspondences for curves. We expect that this theory of Lie algebras associated with curves will unlock a new striking connection between geometry, algebra, and physics, which needs to be investigated further.
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| Outline of Annual Research Achievements |
During the FY2022, I focused on the definition of double (2-dimensional) cohomological Hall algebras (COHAs) and categorified ones (CatHAs) in a general framework. Given a torsion pair (T,F) on the heart of a fixed t-structure of a dg category C (satisfying certain technical conditions), in [arXiv:2207.08926], Diaconescu, Porta, and I introduce a general formalism to associate a COHA (resp. CatHA) to T together with a left and right module (resp. categorified module) associated to F. The full Yangian (resp. categorified Yangian) of T is defined as the monoidal (resp. associative) algebra generated by these two module structures. This framework provides for example a categorification of the Ding-Iohara-Miki algebra of a smooth projective complex surface S introduced by Negut in [arXiv:1703.02027] and a generalization of it including operators of "Hecke modifications along curves"; it allows the definition of actions on the homology, K-theory, etc, of the moduli space of PT stable pairs on S.
In [arXiv:2303.17154], Diaconescu, Porta, and I studied the geometry of the moduli space of PT stable pairs and Hilbert schemes of points on a space curve singularity, using motivic Hall algebras, providing an explicit formula of the generating function of their Euler numbers when the space curve singularity is locally complete intersection.
Finally, Diaconescu, Porta, Schiffmann, Vasserot, and I are finalizing the project about a description of the COHA of the minimal resolution of a type ADE singularity in terms of the affine Yangian of the same type.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The whole project is going as planned.
I made important progress in the part of the project about the double (categorified) Yangian by developing a very general framework: it has been applied to some geometric settings already (notably the case of the COHAs of smooth projective complex surfaces) and I plan to apply to noncompact settings as well (as the case of the cotangent of the curve). In the latter case, the study of the double (categorified) Yangian in the curve case will provide a different viewpoint in the understanding of the Dolbeault COHA of a curve.
The part of the project about Dolbeault COHA in the case of the projective line is almost finished: I aim at finding an explicit description of this COHA by generators and relations, thanks to the relation to affine Yangians.
During my visit to Kavli IPMU in March 2023, I presented it at the conference in honor of the 60th birthday of Hiraku Nakajima. Moreover, I have discussed different aspects of the whole project with the researchers at IPMU and this has been beneficial to the project.
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| Strategy for Future Research Activity |
During FY2023, I plan to pursue further the development of the theory of double categorified Yangians: in particular I aim at computing the categorified commutators in this framework, generalizing the work of Yu Zhao [arXiv:2009.11267, arXiv:2112.12434]. I plan to apply the theory of double categorified Yangians when the surface is the cotangent of a curve, constructing categorified representations in terms of moduli spaces of cyclic Higgs bundles on the curve. Moreover, I plan to study in detail the geometry of moduli spaces of cyclic Higgs bundles on a curve, which are candidates of curve analog of Nakajima quiver varieties. More generally, I plan to study moduli spaces of stable objects on a compactification of the cotangent bundle of a curve, with respect to a fixed Bridgeland stability condition.
In addition, I plan to finish the project about the explicit characterization of the Dolbeault cohomological Hall algebra of the projective line (and, more generally, the cohomological Hall algebra of torsion sheaves on a minimal resolution of a type ADE singularity) via affine Yangians. These cohomological Hall algebras will provide a new positive half of a certain completion of the corresponding affine Yangians.
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